Marc Bogaerts - cp's OEIS frontend

User: Marc Bogaerts

Marc Bogaerts has authored 3 sequences.

A231599 T(n,k) is the coefficient of x^k in Product_{i=1..n} (1-x^i); triangle T(n,k), n >= 0, 0 <= k <= A000217(n), read by rows.

1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 0, 2, 0, 0, -1, -1, 1, 1, -1, -1, 0, 0, 1, 1, 1, -1, -1, -1, 0, 0, 1, 1, -1, 1, -1, -1, 0, 0, 1, 0, 2, 0, -1, -1, -1, -1, 0, 2, 0, 1, 0, 0, -1, -1, 1, 1, -1, -1, 0, 0, 1, 0, 1, 1, 0, -1, -1, -2, 0

Offset: 0

Marc Bogaerts, Nov 11 2013

From Tilman Piesk, Feb 21 2016: (Start)
The sum of each row is 0. The even rows are symmetric; in the odd rows numbers with the same absolute value and opposed signum are symmetric to each other.
The odd rows where n mod 4 = 3 have the central value 0.
The even rows where n mod 4 = 0 have positive central values. They form the sequence A269298 and are also the rows maximal values.
A086376 contains the maximal values of each row, A160089 the maximal absolute values, and A086394 the absolute parts of the minimal values.
Rows of this triangle can be used to efficiently calculate values of A026807.
(End)

			For n=2 the corresponding polynomial is (1-x)*(1-x^2) = 1 -x - x^2 + x^3.
Irregular triangle starts:
  k    0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
n
0      1
1      1  -1
2      1  -1  -1   1
3      1  -1  -1   0   1   1  -1
4      1  -1  -1   0   0   2   0   0  -1  -1   1
5      1  -1  -1   0   0   1   1   1  -1  -1  -1   0   0   1   1  -1

Alois P. Heinz, Rows n = 0..40, flattened
Dorin Andrica and Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11.
Tilman Piesk, Rows n = 0..40 as left-aligned and centered table

Cf. A000217 (triangular numbers).
Cf. A086376, A160089, A086394 (maxima, etc.).
Cf. A269298 (central nonzero values).

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))
        (expand(mul(1-x^i, i=1..n))):
seq(T(n), n=0..10);  # Alois P. Heinz, Dec 22 2013

Table[If[k == 0, 1, Coefficient[Product[(1 - x^i), {i, n}], x^k]], {n, 0, 6}, {k, 0, (n^2 + n)/2}] // Flatten (* Michael De Vlieger, Mar 04 2018 *)

row(n) = pol = prod(i=1, n, 1 - x^i); for (i=0, poldegree(pol), print1(polcoeff(pol, i), ", ")); \\ Michel Marcus, Dec 21 2013

from sympy import poly, symbols
def a231599_row(n):
    if n == 0:
        return [1]
    x = symbols('x')
    p = 1
    for i in range(1, n+1):
        p *= poly(1-x**i)
    p = p.all_coeffs()
    return p[::-1]
# Tilman Piesk, Feb 21 2016

T(n,k) = [x^k] Product_{i=1..n} (1-x^i).

T(n,k) = T(n-1, k) + (-1)^n*T(n-1, n*(n+1)/2-k), n > 1. - Gevorg Hmayakyan, Feb 09 2017 [corrected by Giuliano Cabrele, Mar 02 2018]

A188581 Inverse Moebius transform of A000688, the number of factorizations of n into prime powers greater than 1.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 7, 4, 4, 2, 8, 2, 4, 4, 12, 2, 8, 2, 8, 4, 4, 2, 14, 4, 4, 7, 8, 2, 8, 2, 19, 4, 4, 4, 16, 2, 4, 4, 14, 2, 8, 2, 8, 8, 4, 2, 24, 4, 8, 4, 8, 2, 14, 4, 14, 4, 4, 2, 16, 2, 4, 8, 30, 4, 8, 2, 8, 4, 8, 2, 28, 2, 4, 8, 8, 4, 8, 2, 24, 12, 4, 2, 16, 4, 4, 4, 14, 2, 16

Offset: 1

Marc Bogaerts, Apr 04 2011

			For n=8; the divisors of 8 are 1,2,4,8. There are 1,1,2,3 abelian groups of these orders respectively, so a(n) = 1+1+2+3 = 7.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000688, A000070.

trf:=function ( f, x )  # the Dirichlet convolution 1 * f
    local  d;
    d := DivisorsInt( x );
    return Sum( d, function ( i )
            return f( i );
        end );
end;
nra:=function ( x )     # the number of Abelian Groups of order(n)
    local  pp, ll;
    pp := PrimePowersInt( x );
    ll := [ 1 .. Size( pp ) / 2 ];
    return Product( List( 2 * ll, function ( i )
              return NrPartitions( pp[i] );
          end ) );
end;
a:=function ( n )
    return trf( nra, n );
end;

with(combinat): with(numtheory):
a:= n-> add(mul(numbpart(i[2]), i=ifactors(d)[2]), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 08 2011

InverseMobiusTransform[a_List] := Module[{n = Length[a], b}, b = Table[0, {i, n}]; Do[b[[i]] = Plus @@ a[[Divisors[i]]], {i, n}]; b]; A688[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger@n; InverseMobiusTransform[Array[A688, 100]] (* T. D. Noe, Apr 07 2011 *)
f[0] = 1; f[e_] := f[e] = f[e - 1] + PartitionsP[e]; a[1] = 1; a[n_] := Times @@ (f[Last[#]] & /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 09 2020 *)

A000688(n)={local(f); f=factor(n); prod(i=1, matsize(f)[1], numbpart(f[i, 2]))}
A188581(n)=sumdiv(n,d,A000688(d))
r=vector(66,n,A188581(n)) /* show terms */ /* Joerg Arndt, Apr 08 2011 */

a(n) = Sum_{d | n} A000688(d).
Multiplicative with a(p^e) = A000070(e). - Amiram Eldar, Sep 09 2020
Dirichlet g.f.: zeta(s)^2 * Product_{k>=2} zeta(k*s). - Ilya Gutkovskiy, Nov 03 2020
Sum_{k=1..n} a(k) ~ n*((log(n) + 2*gamma - 1)*f(1) + f'(1)), where f(1) = Product_{k>=2} zeta(k) = A021002 = 2.1955691982567064617939..., f'(1) = f(1) * Sum_{k>=2} k*zeta'(k)/zeta(k) = -5.0385164470942955610707128990779476296197... and gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Aug 21 2021

A188585 Moebius inversion of sequence A000688, the number of factorizations of n into prime powers greater than 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

Marc Bogaerts, Apr 04 2011

Dirichlet convolution product of A000688 with the Moebius function.
It appears that a(n) is nonzero for n in A001694, the powerful numbers. - T. D. Noe, Apr 06 2011 [This is correct: a(n) > 0 if and only if n is in A001694. - Amiram Eldar, Jun 10 2025]
There is a similar sequence defined by b(n) = Product_{i} floor(e(i)/2) where n = Product_{p} p(i)^e(i) is the usual prime factorization, which differs from a(n) at n = 64, 128, 256, 512, 576, 729,.... - R. J. Mathar, Sep 18 2012 [This sequence is A365550. - Amiram Eldar, Jun 10 2025]
The number of unordered factorizations of n into 1 and prime powers p^e where p is prime and e >= 2 (A025475). - Amiram Eldar, Jun 10 2025

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000688, A001694, A002865, A008683, A025475, A365550.

mtrf:=function ( f, x )     # the Moebius inversion formula
    local  d;
    d := DivisorsInt( x );
    return Sum( d, function ( i )
            return f( i ) * MoebiusMu( (x / i) );
        end );
end;
nra:=function ( x )         # the number of Abelian groups of order x
    local  pp, ll;
    pp := PrimePowersInt( x );
    ll := [ 1 .. Size( pp ) / 2 ];
    return Product( List( 2 * ll, function ( i )
              return NrPartitions( pp[i] );
          end ) );
end;
a:=function ( n )
    return mtrf( nra, n );
end;

with(numtheory): with(combinat):
a:= n-> add(mobius(n/d) *mul(numbpart(i[2]),
        i=ifactors(d)[2]), d=divisors(n)):
seq(a(n), n=1..110);  # Alois P. Heinz, Apr 07 2011

MobiusTransform[a_List] := Module[{n = Length[a], b}, b = Table[0, {i, n}];Do[b[[i]] = Plus @@ (MoebiusMu[i/Divisors[i]] a[[Divisors[i]]]), {i, n}]; b]; A688[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger@n; MobiusTransform[Array[A688, 100]] (* T. D. Noe, Apr 06 2011 *)
f[p_, e_] := PartitionsP[e] - PartitionsP[e-1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 10 2025 *)

a(n) = vecprod(apply(x -> numbpart(x)-numbpart(x-1), factor(n)[, 2])); \\ Amiram Eldar, Jun 10 2025

from math import prod
from sympy import partition, factorint
def A188585(n): return prod(partition(e)-partition(e-1) for e in factorint(n).values()) # Chai Wah Wu, Jun 10 2025

a(n) = Sum_{d|n} A008683(n/d) * A000688(d).
Dirichlet g.f.: Product_{k>=2} zeta(k*s). - Ilya Gutkovskiy, Nov 03 2020
Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = Product_{k>=3} zeta(k/2) = 10.0301441966843566206076085895839492473559217336... - Vaclav Kotesovec, Apr 22 2025
Multiplicative with a(p^e) = A002865(e). - Amiram Eldar, Jun 10 2025

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User: Marc Bogaerts

A231599 T(n,k) is the coefficient of x^k in Product_{i=1..n} (1-x^i); triangle T(n,k), n >= 0, 0 <= k <= A000217(n), read by rows.

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A188581 Inverse Moebius transform of A000688, the number of factorizations of n into prime powers greater than 1.

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Author

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Maple

Mathematica

PARI

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A188585 Moebius inversion of sequence A000688, the number of factorizations of n into prime powers greater than 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Python

Formula